The Bohr Model of the Hydrogen Atom — Question 5

Question 5

Reduced nuclear mass predicts an isotope shift without changing the level indices

For a one-electron atom with nuclear mass M, replace $m_e$ by reduced mass $\mu=m_eM/(m_e+M)$. At fixed Z, each transition frequency is proportional to $\mu$. Model hydrogen by $M_H=1836m_e$ and deuterium by $M_D=3672m_e$.

1. Find $\nu_D/\nu_H$ for the same transition.

2. Find the fractional wavelength decrease $(\lambda_H-\lambda_D)/\lambda_H$.

Question 5 - Solution

Reduced nuclear mass predicts an isotope shift without changing the level indices

(a) Method and calculation

$$\mu_H/m_e=1836/1837,\qquad\mu_D/m_e=3672/3673.$$$$\boxed{\nu_D/\nu_H=\frac{3672}{3673}\frac{1837}{1836}=\frac{3674}{3673}}.$$

(b) Interpretation and further calculation

$$\lambda_D/\lambda_H=\nu_H/\nu_D=3673/3674.$$$$\boxed{(\lambda_H-\lambda_D)/\lambda_H=1/3674=2.72183\times10^{-4}}.$$

Physical check

The heavier isotope has a reduced mass closer to the electron mass, increasing transition energies and decreasing wavelengths. This model excludes other isotope-dependent corrections.